本篇論文中，我們希望透過次波長光柵結構來設計出低損耗、低跨接波導耦合能量、小尺寸的跨接元件。
1.我們討論次波長光柵結構在應用上的限制與原理。
2.回顧目前最常見的跨接波導元件設計，並探討其中優缺點。
3.利用次波長光柵結構來達到我們設計波導跨接元件的目標。
4.透過三維有限時域差分(3D FDTD)法來模擬設計元件在使用上的結果與分析。
經由模擬結果，我們利用寬的次波長光柵波導，可以設計出跨接損耗約0.13dB
且跨接波導耦合能量能夠達到<-35dB的元件。
然而在尺寸方面，因為次波長光柵本身的限制，必須超過20個週期的數量，使得我們的元件尺寸為12 x 12 μm^2。

In this thesis, we try to design a low-loss, low- crosstalk and compact dimension waveguide crossing structure with subwavelength grating structures.
1.We discuss the theory and the application limits of subwavelength grating (SWG) structures.
2.Review SOI waveguide crossing designs and discuss the strengths and weakness .
3.Design waveguide crossing structure with subwavelength structures.
4.Discuss and analysis the results from 3D FDTD simulation.
According to the simulation results, we can get a waveguide crossing structure with 0.13dB loss and crosstalk <-35dB.
But the crossing dimension would be limited by the characteristics of SWGs, we need more than 20 period number to get efficient SWGs. The crossing dimension is 12 x 12 μm^2 of our design.

論文審定書 i
致謝 iii
中文摘要 iv
Abstract v
List of Figures viii
List of Tables xii
Chapter 1: Introduction 1
1.1 Background 1
1.2 Motivation 2
1.3 Structures of thesis 4
Chapter 2: Principle of subwavelength grating structures 5
2.1 Bloch-Floquet theory 5
2.2 Effective medium theory 7
2.3 Application of Subwavelength Grating 8
2.3.1 Concept of subwavelength grating 8
2.3.2 The choice of subwavelength grating period 10
2.3.3 Effective index and filling factor 13
Chapter 3: Literature review of SOI waveguide crossing design 16
3.1 Multimeode interference based waveguide crossing design 16
3.1.1 Principle of multimode interference 16
3.1.2 Device structure 17
3.2 Mode expansion tapers based waveguide crossing design 19
3.2.1 Concept of mode expansion taper 19
3.2.2 One-step etching process for waveguide crossing 19
3.2.3 Two-step etching process for waveguide crossing 21
3.3 Subwavelength gratings based waveguide crossing design 23
3.3.1 Design and results 23
3.3.2 Steady state mode and period number 25
3.3.3 Bridge structures for Reflection 28
Chapter 4: Subwavelength grating crossing Design and Analysis 30
4.1 SWG crossing design based on effective medium theory 30
4.1.1 Taper design 30
4.1.2 Cross center design 34
4.1.3 Results and analysis 35
4.2 Crossing design based on subwavelength gratings 39
4.2.1 Taper Design 39
4.2.2 Cross Center Design 44
4.2.3 Results and analysis 46
Chapter 5: Conclusion 56
Reference 58
Appendix A Finite-difference time-domain 62

[1]Pablo Sanchis, Pablo Villalba, Francisco Cuesta, Andreas Håkansson, Amadeu Griol, José V. Galán, Antoine Brimont, and Javier Martí, “Highly efficient crossing structure for silicon-on-insulator waveguides,” Optics Letters, Vol. 34, Issue 18, pp. 2760-2762 (2009)
[2] Y. Zhang, S. Yang, A. E.-J. Lim, G.-Q. Lo, C. Galland, T. Baehr Jones, M. Hochberg, "A CMOS-compatible, low-loss, and low-crosstalk silicon waveguide crossing ," IEEE Photonics Technology Letters, vol. 25, issue 5, pp. 422-425(2013)
[3] Wim Bogaerts, Pieter Dumon, Dries Van Thourhout, and Roel Baets “Low-loss, low-cross-talk crossings for silicon-on-insulator nanophotonic waveguides,” Optics Letters, Vol. 32, Issue 19, pp. 2801-2803 (2007)
[4] Yoshinori Watanabe, Yoshimasa Sugimoto, Naoki Ikeda, Nobuhiko Ozaki, Akio Mizutani, Yoshiaki Takata, Yoshinori Kitagawa, and Kiyoshi Asakawa,“Broadband waveguide intersection with low crosstalk in two-dimensional photonic crystal circuits by using topology optimization,” Optics Express, Vol. 14, Issue 20, pp. 9502-9507 (2006)
[5] Przemek J. Bock, Pavel Cheben, Jens H. Schmid, Jean Lapointe, André Delâge, Dan-Xia Xu, Siegfried Janz, Adam Densmore, and Trevor J. Hall “Subwavelength grating crossings for silicon wire waveguides” Optics Express, Vol. 18, Issue 15, pp. 16146-16155 (2010)
[6] Steven G. Johnson, Christina Manolatou, Shanhui Fan, Pierre R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross talk in waveguide intersections,” Optics Letters, Vol. 23, Issue 23, pp. 1855-1857 (1998)
[7] CH Chen, CH Chiu “Taper-integrated multimode-interference based waveguide crossing design,” IEEE Journal of Quantum Electronics, vol. 46, issue 11, pp. 1656-1661
[8] S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP NO.2, pp.466-475(1956).
[9] Philippe Lalanne and Jean-Paul Hugonin, “High-order effective-medium theory of subwavelength gratings in classical mounting: application to volume holograms,” JOSA A, Vol. 15, Issue 7, pp. 1843-1851 (1998)
[10] Lalanne, Philippe, and Mike Hutley. "The optical properties of artificial media structured at a subwavelength scale." Encyclopedia of optical engineering: pp.62-71 (2003).
[11] J. Favard, “Sur les équations diff´erentielles linéaires à coefficients périodiques,” Journal Acta Mathematica Vol. 51, Issue 1 , pp 31-81(1928)
[12] F. Bloch, “Über die quantenmechanik der electronen in kristallgittern,” Zeitschrift für Physik Vol. 52, Issue 7-8 , pp 555-600(1929)
[13] Joannopoulos, John D.; Johnson, Steven G.; Winn, Joshua N.; Meade, Robert D. “Photonic crystals molding the flow of light,” Princeton University Press(2008).
[14] D. J. Bergman and K.-J. Dunn, “Bulk effective dielectric constant of a composite with a periodic microgeometry,” Physical Review B (Condensed Matter), Vol. 45, Issue 23, pp.13262-13271(1992)
[15] S. Datta, C. T. Chan, K. M. Ho and C. M. Soukoulis, “Effective dialectic constant of periodic composite structures,” Physical Review B (Condensed Matter), Vol. 48, Issue 20, pp.14936-14943(1993)
[16] Bock, Przemek J. ,Hall, Trevor J. “Addressing the challenges of silicon waveguides with subwavelength nanostructures,” Diss. University of Ottawa(2011).
[17] Bock PJ, Cheben P, Schmid JH, Lapointe J, Delâge A, Janz S, Aers GC, Xu DX, Densmore A, Hall TJ. “Subwavelength grating periodic structures in silicon-on-insulator: a new type of microphotonic waveguide” Optics Express, vol. 18, issue 19, p. 20251(2010)
[18] K. Okamoto, “Fundamentals of Optical Waveguides,” 2nd ed. Academic Press, NY, 2006.
[19] L. B. Soldano and E. C. M. Pennings, “Optical Multi-Mode Interference Devices Based on Self-Imaging：Principles and Applications” Journal of Lightwave Technology, vol. 13, no. 4, pp.615-627, (1995)
[20] M. Bachmann, P. A. Besse, and H. Melchior, “Overlapping-image multimode interference couplers with a reduced number of self-images for uniform and nonuniform power splitting,” Applied Optics, Vol. 34, Issue 30, pp. 6898-6910 (1995)
[21] A. Maese-Novo, R. Halir, S. Romero-García, D. Pérez-Galacho, L. Zavargo-Peche, A. Ortega-Moñux, I. Molina-Fernández, J. G. Wangüemert-Pérez, and P. Cheben
“Wavelength independent multimode interference coupler” Optics Express, Vol. 21, Issue 6, pp. 7033-7040 (2013)
[22] Chen, Hui; Poon, Andrew W.“Low-loss multimode-interference-based crossings for silicon wire waveguides” IEEE Photonics Technology Letters, vol. 18, issue 21, pp. 2260-2262(2006)
[23] Liu, Heliang; Tam, Hwayaw; Wai, P. K. A.; Pun, Edwin “Low-loss waveguide crossing using a multimode interference structure” Optics Communications, Volume 241, Issue 1-3, p. 99-104.(2004)
[24] Fang Xu; Andrew W. Poon “Silicon cross-connect filters using microring resonator coupled multimode-interference-based waveguide crossings” Optics Express, Vol. 16, Issue 12, pp. 8649-8657 (2008)
[25] Chiu, Chia-Hsiang; Chiu, Chia-Hsiang“Taper-integrated multimode-interference based waveguide crossing design” IEEE Journal of Quantum Electronics, vol. 46, issue 11, pp. 1656-1661(2010)
[26] Burns, W. K.; Milton, A. F.; Lee, A. B. “Optical waveguide parabolic coupling horns” Applied Physics Letters, vol. 30, Jan. 1, , p. 28-30.( 1977)
[27] Fukazawa, Tatsuhiko, Hirano, Tomohisa; Ohno, Fumiaki; Baba, Toshihiko “ Low loss intersection of Si photonic wire waveguides” Japanese Journal of Applied Physics, Vol. 43, Issue 2, pp. 646 (2004).
[28] H.-W. Chang, Y.-H. Wu, and W.-C. Cheng “Hybrid FD-FD analysis of crossing waveguides by exploiting both the plus and the cross structural symmetry” Progress In Electromagnetics Research, Vol. 103, 217-240(2010).
[29] Yee, Kane “Numerical solution of initial boundary value problems involvingMaxwell’s equations in isotropic media” IEEE Transactions on Antennas and Propagation, vol. 14, issue 3, pp. 302-307(1966)
[30] K.C Lin“On the characteristic of ring couplers design” M.S. Thesis, University NSYS, July. 2012
[31] Berenger, J.P. “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves” Journal of Computational Physic, Vol. 114, Issue 2 ,pp 185-200(1994)